Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Petersen graph
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== Embeddings == The Petersen graph is [[planar graph|nonplanar]]. Any nonplanar graph has as [[minor (graph theory)|minor]]s either the [[complete graph]] <math>K_5</math>, or the [[complete bipartite graph]] <math>K_{3,3}</math>, but the Petersen graph has both as minors. The <math>K_5</math> minor can be formed by contracting the edges of a [[perfect matching]], for instance the five short edges in the first picture. The <math>K_{3,3}</math> minor can be formed by deleting one vertex (for instance the central vertex of the 3-symmetric drawing) and contracting an edge incident to each neighbor of the deleted vertex. [[Image:Petersen graph, two crossings.svg|class=skin-invert-image|thumb|right|The Petersen graph has [[Crossing number (graph theory)|crossing number]] 2 and is [[1-planar graph|1-planar]].<ref>{{citation | last = Loupekine | first = Feodor | doi = 10.21954/OU.RO.0000E032 | publisher = The Open University | type = Ph.D. thesis | title = Approaches to the four colour theorem | url = https://oro.open.ac.uk/id/eprint/57394 | date = October 1992}}; see Figure 3.4, p. 28</ref>]] The most common and symmetric plane drawing of the Petersen graph, as a pentagram within a pentagon, has five crossings. However, this is not the best drawing for minimizing crossings; there exists another drawing (shown in the figure) with only two crossings. Because it is nonplanar, it has at least one crossing in any drawing, and if a crossing edge is removed from any drawing it remains nonplanar and has another crossing; therefore, its crossing number is exactly 2. Each edge in this drawing is crossed at most once, so the Petersen graph is [[1-planar graph|1-planar]]. On a [[torus]] the Petersen graph can be drawn without edge crossings; it therefore has [[genus (mathematics)|orientable genus]] 1. [[Image:Petersen graph, unit distance.svg|class=skin-invert-image|thumb|right|The Petersen graph is a [[unit distance graph]]: it can be drawn in the plane with each edge having unit length.]] The Petersen graph can also be drawn (with crossings) in the plane in such a way that all the edges have equal length. That is, it is a [[unit distance graph]]. The simplest [[surface (mathematics)|non-orientable surface]] on which the Petersen graph can be embedded without crossings is the [[projective plane]]. This is the embedding given by the [[hemi-dodecahedron]] construction of the Petersen graph (shown in the figure). The projective plane embedding can also be formed from the standard pentagonal drawing of the Petersen graph by placing a [[cross-cap]] within the five-point star at the center of the drawing, and routing the star edges through this cross-cap; the resulting drawing has six pentagonal faces. This construction forms a [[Regular map (graph theory)|regular map]] and shows that the Petersen graph has [[genus (mathematics)|non-orientable genus]] 1. [[File:Petersen-graph.png|thumb|The Petersen graph and associated map embedded in the [[projective plane]]. Opposite points on the circle are identified, yielding a closed surface of non-orientable genus 1.]]
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)